The volume of a sphere with radius r is which expression?

• A. 2πr^3
• B. 4πr^3
• C. 4/3 π r^3
• D. 1/3 π r^2

Answer: (C)

Think about building the sphere from very thin disks stacked along a diameter. At a distance x from the center, a slice has radius sqrt(r^2 − x^2), so its area is π(r^2 − x^2). Adding up all the slices from x = −r to x = r means integrating: V = ∫ from −r to r of π(r^2 − x^2) dx. That integral works out to π[r^2x − x^3/3] from −r to r, which equals (4/3)πr^3. The volume must scale with the cube of the radius, and the correct coefficient is 4/3. The other expressions either have the wrong power of r or the wrong coefficient (or are missing the r^3 term entirely).